Luminous compact blue galaxies (LCBGs) have enhanced star formation rates and compact morphologies. We combine Sloan Digital Sky Survey data with HI data of 29 LCBGs at redshift z~0 to understand their nature. We find that local LCBGs have high atomic gas fractions (~50%) and star formation rates per stellar mass consistent with some high redshift star forming galaxies. Many local LCBGs also have clumpy morphologies, with clumps distributed across their disks. Although rare, these galaxies appear to be similar to the clumpy star forming galaxies commonly observed at z~1-3. Local LCBGs separate into three groups: 1. Interacting galaxies (~20%); 2. Clumpy spirals (~40%); 3. Non-clumpy, non-spirals with regular shapes and smaller effective radii and stellar masses (~40%). It seems that the method of building up a high gas fraction, which then triggers star formation, is not the same for all local LCBGs. This may lead to a dichotomy in galaxy characteristics. We consider possible gas delivery scenarios and suggest that clumpy spirals, preferentially located in clusters and with companions, are smoothly accreting gas from tidally disrupted companions and/or intracluster gas enriched by stripped satellites. Conversely, as non-clumpy galaxies are preferentially located in the field and tend to be isolated, we suggest clumpy, cold streams, which destroy galaxy disks and prevent clump formation, as a likely gas delivery mechanism for these systems. Other possibilities include smooth cold streams, a series of minor mergers, or major interactions.